The first theorem shows that the TV norm-penalized problem (REF ) is equivalent to the consensus-constrained one (REF ) (and hence (REF ) too), when the penalty parameter \(\lambda \)  is sufficiently large. This theorem is analogous to Theorem 1 in [1]}, but our proof is based on a system of linear equations involving the decentralized network structure and different from the proof of Theorem 1 in [1]}.
We define \(A \in \mathbb {R}^{|\mathcal {R}| \times |\mathcal {E}_{R}|}\)  as the node-edge incidence matrix of \((\mathcal {R},\mathcal {E}_{R})\) . To be specific, for an edge \(e=(i,j) \in \mathcal {E}_{R}\)  with \(i<j\) , the \((i,e)\) -th entry of \(A\)  is 1 while the \((j,e)\) -th entry of \(A\)  is \(-1\) .
